Error distributions, bond distances

For the one-electron and //-electron hierarchies of ab initio theory to be useful, it is necessary to carry out a careful and extensive calibration of their performance. Such a calibration is best carried out by comparing, in a statistical manner, calculated values of different properties with experimental measurements. In the present section, we give two examples of such comparisons, for bond lengths and reaction enthalpies. In Figs. 6 and 7, we have plotted the normal distributions of the errors in calculations of bond distances and reaction enthalpies, respectively. The statistics underlying the normal distributions are based on calculations on 19 closed-shell first-row molecules (e.g., HF and C2H4) and 13 reactions involving these molecules (for more details, see Refs. 1 and 17). 

Figure 6 Normal distributions of the error in calculated bond distances (pm).

In the largest basis (cc-pCVQZ), the distribution of errors is very broad for Hartree-Fock but becomes sharply peaked at the CCSD(T) level. The MP2 and CCSD distributions are similar to each other and intermediate between those of Hartree-Fock and CCSD(T). It is important to note, however, that this progression of the n-electron models is not observed in the small cc-pCVDZ basis. Clearly, this basis is not sufficiently flexible for correlated calculations, providing too small virtual excitation spaces for these models to work properly—in particular, for reaction enthalpies. In the cc-pCVTZ basis, convergence is more satisfactory, especially for bond distances. 

Figure 5. H -O frequency distributions in crystals for hydrogen bonds from C=C-H donors to (top) C=0 acceptors, n = 33, and (bottom) C-OH acceptors, n = 31. Database analysis perform for this article [CSD, June 1997 update with 167797 entries, ordered and error-free crystal structures with R < 0.10, normalized H-atom position based on a linear C=C-H group, and a C-H bond distance of 1.08 A neither donor nor acceptor group directly bonded to a metal atom, H -O < 2.95 A for three-center hydrogen bonds, only the short component is considered).

The atoms and bonds are positioned sequentially, by trial and error, such that the bond distribution of an added atom meshes with its already-placed neighbors. When a ring cannot be properly closed, or an atom or bond cannot be placed without crowding another atom already placed, then backtracking is required. The first tactic taken is to increase or decrease the bond distance at the last decision point. If this fails for all decision points, a different distribution is tried from step 5. If failure persists, control is returned to step 4 with a larger margin. 

Having discussed the mean errors in the bond distances, it is appropriate to consider the standard deviations so as to characterize more fully the distribution of errors in the calculations. Only for MP2, MP4 and CCSD(T) does the standard deviation decrease monotonically with improvements in the basis - see Figure 15.3. For MP3 and CCSD, it decreases from cc-pVDZ to cc-pVTZ but then increases as we go to cc-pVQZ. For the CISD model, the standard deviation increases monotonically for the Hartree-Fock model, it is always large. 

In Figure 15.4, we have, for each basis and each A-electron model, plotted the normal distributions for the calculated bond distances. Although we make no claim that the errors in the calculated bond distances are indeed normally distributed, these plots neatly summarize the performance of the various levels of theory. 

Fig. 15.4. Normal distributions of the errors in the calculated bond distances (pm). For ease of comparison, all distributions have been normalized to one and plotted against the same horizontal and vertical scales.

In Figure 15.7, we have plotted the normal distributions of the errors in the calculated bond angles. The main difference from the plots for bond distances in Figure 15.4 is that, for bond angles, the CCSD(T) distributions are less sharply peaked and similar to those of the MP2 model, reflecting the poorer quality of the experimental reference data for bond angles than for bond distances. 

Hesselink (1969) has argued that Meier s derivation for tails incorporates a procedural error that invalidates his quantitative results for the mixing free energy. Meier formulated a probability distribution function WJ x,d), which describes the probability that the terminal (A th) bond of the tail lies at a distance in th ange xiox+dx from the surface when the plate separation is d. The segment density distribution function was then evaluated by summing the end-to-end probability functions for all subchains of k bonds over the total number of bonds n... 

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